Magnetic Properties of Solids

Materials may be classified by their response to externally applied magnetic fields as diamagnetic, paramagnetic, or ferromagnetic. These magnetic responses differ g p g g g pgreatly in strength. Diamagnetism is a property of all materials and opposes applied magnetic fields, but is very weak. Paramagnetism, when present, is stronger than diamagnetism and produces magnetization in the direction of the applied field, and proportional to the applied field. Ferromagnetic effects are very large; producing magnetizations sometimes orders of magnitude greater than the applied field and as such are much larger than either diamagnetic or paramagnetic effects. The magnetization of a material is expressed in terms of density of net magnetic dipole moments m in the material. We define a vector quantity called the

ti ti M b M /V Th th t t l ti fi ld B i th t i l imagnetization M by M = μtotal/V. Then the total magnetic field B in the material is given by B = B0 + μ0M where μ0 is the magnetic permeability of space and B0 is the externally applied magnetic field. When magnetic fields inside of materials are calculated using Ampere's law or the Biot Savart law then the μ in thosecalculated using Ampere s law or the Biot-Savart law, then the μ0 in those equations is typically replaced by just μ with the definition μ = Kmμ0 where Km is called the relative permeability. If the material does not respond to the external magnetic field by producing any magnetization then K = 1magnetic field by producing any magnetization, then Km = 1.

Magnetic Properties of SolidsAnother commonly used magnetic quantity is the magnetic susceptibility which specifies how much the relative permeability differs from one. Magnetic susceptibility χ K 1 For paramagnetic and diamagnetic materials thesusceptibility χm = Km – 1 For paramagnetic and diamagnetic materials the relative permeability is very close to 1 and the magnetic susceptibility very close to zero. For ferromagnetic materials, these quantities may be very large. Another way to deal with the magnetic fields which arise from magnetization ofAnother way to deal with the magnetic fields which arise from magnetization of materials is to introduce a quantity called magnetic field strength H . It can be defined by the relationship H = B0/μ0 = B/μ0 - M and has the value of unambiguously designating the driving magnetic influence from externalunambiguously designating the driving magnetic influence from external currents in a material, independent of the material's magnetic response. The relationship for B above can be written in the equivalent form B = μ0(H + M) H and M will have the same units, amperes/meter. Ferromagnetic materials willH and M will have the same units, amperes/meter. Ferromagnetic materials will undergo a small mechanical change when magnetic fields are applied, either expanding or contracting slightly. This effect is called magnetostriction.

Diamagnetism

The orbital motion of electrons creates tiny atomic current loops, which produce magnetic fields. When an external magnetic field is applied to a p oduce ag et c e ds e a e te a ag et c e d s app ed to amaterial, these current loops will tend to align in such a way as to oppose the applied field. This may be viewed as an atomic version of Lenz's law: induced magnetic fields tend to oppose the change which created them.induced magnetic fields tend to oppose the change which created them. Materials in which this effect is the only magnetic response are called diamagnetic. All materials are inherently diamagnetic, but if the atoms have some net magnetic moment as in paramagnetic materials, or ifhave some net magnetic moment as in paramagnetic materials, or if there is long-range ordering of atomic magnetic moments as in ferromagnetic materials, these stronger effects are always dominant. Diamagnetism is the residual magnetic behavior when materials areDiamagnetism is the residual magnetic behavior when materials are neither paramagnetic nor ferromagnetic. Any conductor will show a strong diamagnetic effect in the presence of changing magnetic fields because circulating currents will be generatedchanging magnetic fields because circulating currents will be generated in the conductor to oppose the magnetic field changes. A superconductorwill be a perfect diamagnet since there is no resistance to the forming of the current loopsthe current loops.

Paramagnetism

Some materials exhibit a magnetization which is proportional to the applied g p p ppmagnetic field in which the material is placed. These materials are said to be paramagnetic and follow Curie's law:

⎞⎛ B

KelvinsineTemperaturTConstantsCurie'C

field Magnetic Bion MagnetizatM;

==

==⎟⎠⎞

⎜⎝⎛=TBCM

All atoms have inherent sources of magnetism because electron spin contributes a magnetic moment and electron orbits act as current loops which produce a

ti fi ld I t t i l th ti t f th l t l b t

Kelvinsin eTemperaturT Constant sCurieC ==

magnetic field. In most materials the magnetic moments of the electrons cancel, but in materials which are classified as paramagnetic, the cancellation is incomplete.

Magnetostriction

Magnetostriction

It is also observed that applied mechanical strain produces some magnetic anisotropy. If an iron crystal is placed under tensile stress, then the direction of the stress becomes the preferred magnetic direction and the domains will tend to line up in that direction. Ordinarily the direction of magnetization in iron is easily changed by rotating the applied magnetic field, but if there is tensile stress in the iron sample, there is some resistance to that rotation of direction. Bulk solid samples may have internal strains which influence the domain boundary movement. M t t i ti b d t t ib t h ll lMagnetostriction can be used to create vibrators, where usually some lever action is used in conjunction with the magnetic deformation to increase the resultant amplitude of vibration. Magnetostriction is also used to produce

lt i ib ti ith d lt i iultrasonic vibrations either as a sound source or as ultrasonic waves in liquids which can act as a cleaning mechanism in ultrasonic cleaning devices.

Hysteresis Curvesy

Properties of Permalloy thin filmsProperties of Permalloy thin films Ms=10/4π kG Hc=0.3 Oe Hk= 5 OeApplications: computer memory, magnetoresistance, detector, reading Heads

Magnetic Energiesg g• Exchange energy

alignment of spins, cost of energy to change direction ofenergy to change direction of magnetizationcompensated by thermal energy ⇒ phase transition at T

JS≈ 2

2

exchangeσ⇒ phase transition at Tc• Magnetostatic energy

discontinuity of normal component across interface

Na2exchange

across interface⇒ demagnetizing factor f(sample shape)

• Magnetocrystalline anisotropypreference of magnetization along KNa≈i tσpreference of magnetization along crystallographic directions

• Magnetoelastic energyh f ti ti d t t i

KNa≈anisotropyσQuantities:J = exchange integralchange of magnetization due to strain

(magnetostriction)•Zeeman energy

t ti l f ti t i

J = exchange integralS = spina = atomic distanceN = number of spinspotential energy of magnetic moment in

a fieldN = number of spinsK = anisotropy constant

Stoner-Wohlfarth model

Free energy in magnetic anisotropygy g py

02

1 sin ϕKE =K1 = uniaxial anisotropy

l b t M d i

01 ϕ

φ0= angle between M and easy axisEA easy axis for energy minimaHA hard axis for energy maximaHA hard axis for energy maxima

Single Domain particlesg pFerromagnetic particles sufficiently small

z z

EAx EAx

Condition 1

HAh

M

EAβ φ0 EAβ φ0

HA

EAβφ0

)cos(sin 002

1 −−= ϕβϕ HMKE

Applying external field H;

2,sin,cos,,

parameters new define

1|| ≡==≡≡ ⊥

ββεMKH

HHh

HHh

HHh

MHE

KKKKK

sincossin

field anisotropy

00||02

21

∂

−−=

=

⊥

ε

ϕϕϕε hh

HK

0Mon ;net torque

0extrema

0

0

∂=Λ

=∂∂

⇒

ε

ϕε

Condition 1

0 ifonly (stable)answer

0cossin2sin

2

2

0

00||021

00

>∂∂

=

=−+=∂∂

=Λ ⊥

εϕ

ϕϕϕϕε hh

sincos2cos

y( )

100||00

020

2

20

0

Λ=++=∂Λ∂

=∂∂

∂

⊥ ϕϕϕϕϕ

ε

ϕϕ

hh

andEA|| H: 1condition

effective uniform1 =Λ

hyteresis noEA H:2 condition ⇒⊥

Condition 2

Various magnetic anisotropy energies

Shape anisotropy energyShape anisotropy energya measure of the difference in the energies associated with magnetization in the shortest and longest di i f f ti b ddimensions of a ferromagnetic body

Magnetocrystalline anisotropySt i m g tost i tio iost opStrain-magnetostriction aniostropyM-induced uniaxial anisotropy

bl dOblique incident anisotropy

Magnetostatic Energy

Large MS energy

Smaller MS energy

Smaller MShigher

wall energy

No MSenergy

Closure domains: in magnetic hard directions problem: magnetostriction!

Domain wall Energy

⎟⎞

⎜⎛ KkTAK c 14)(4γ ⎟

⎠⎜⎝

==a

AK c 11 4)(4γ

Domain wall width

a=lattice spacing, Tc=curie temperature, k = Boltzmann constant

⎟⎟⎠

⎞⎜⎜⎝

⎛==

11

)(aKkT

KA cππδ

⎠⎝ 11

Domain Wall EnergygyEnergetic considerations:domain wall costs wall energy but reduces magnetostatic energydomain wall costs wall energy, but reduces magnetostatic energy

More Domains = smaller spacing d

↑Magnetostatic energy density ↑Domain wall energy density ↓

Thin films are frequently single domain, magnetization in-plane

Domain Wall Energygy

Intrinsic magnetic properties (approximate values) of a typical hard magnetic materials (SmCo5) and a typical soft magnetic material (Fe)

Ref: R.A. McCurrie Ferromagnetic materials : structure and properties, Academic Press, 1994,Table 1.3

Domain WallJSF ijij =

2 )cos(2 θ

JS 2

energy exchange

NaJS

≈ 2exchange

energyAnisotropy

σ

KNa≈anisotropy

energy Anisotropyσ

Quantities:J = exchange integralS = spina = atomic distanceN = number of spinsK = anisotropy constant

Types of Domain Wallsyp

Bloch and Néel Walls

out of planein-plane

Cross-Tie Wall

Magnetocrystalline Anisotropy

Magnetic Films / Size Effectsg zsingle domain particles:ffirst approximation:particle size ~ domain wall size → no domain walls single domain particles→ no domain walls, single domain particlesmore detialed: include magneto static energygytypical rc = 3nm (Fe)

Size effectsz

Superparamagnetism:small particles:magnetic direction is not fixed by anisotropy or shapemagnetic direction is not fixed by anisotropy or shapethermal energy might change / flip magnetic momentrsp = 20 nmrsp 20 nmeach particle ferromagnet, but particles disordered=> behavior like paramagnet, but higher permeabilityhigh Ms, but no Hc

Size Effects: Summaryz y

Magnetism in Thin Films/Small structures

In/Out of plane magnetization

Stress and Magnetization Ig z

Stress and Magnetization II

Exchange Energy Couplingg gy p g

Giant Magnetoresistanceg

GMR (Fe/Cr Multilayer)y

GMR: Theory and explanation

Equivalent circuit

Spin Valvep

Link to animation

Application: Data storagepp g

Requirementsq

Recording mediumg

Criteria for magnetic properties

Noise

Longitudinal vs Perpendicular recording

Particulate Recording media

Single domain particlesAcicular particles due to shape anisotropyAcicular particles due to shape anisotropyembedded in polymer matrixalignments by suitable deposition process or baking in

ti fi ldmagnetic field

not for high density media (Bit length > 1μm)total magnetization reduced due to binderapplications, tape, floppy diskpp , p , ppymaterials: CrO2, γ-Fe2O3Co doped γ-Fe2O3 to improve coercitivity, either alloyed or surface layersurface layerpure iron + oxidation/corrosion protection

Characteristics of Particulate media

Thin film recording media

Inductive Recording Media

Various materials for inductive recording heads

Write Head

Signal Strength in read head

)sin(∝ − kxeH kxx

901)exp(

0 =−=−

= −∫

edyky

V kd

d

d

9.01)exp(

0

==−

= ∞∞

∫e

dykyV

λ37.090 ≈D

Magnetoresistive headg

Magnetoresistive headMagnetoresitance = magnetic field changes electrical resistivity

g

different mechanisms possible!

Anisotropic magnetic resistance (AMR) 2-3%p g ( ) %magnetic field causes oscillation of conduction electronsGiant magneto resistance (GMR) up to 100 %Magnetic field changes alignement of antiferromagnetic layersCollossal magneto resistance (CMR) several 100 %Magnetic field induces phase transition ceramic insulator® metalMagnetic field induces phase transition ceramic insulator® metal

GMR and spinvalvesp

Summary: requirementsy qwrite head:

read head:•Low coercivity

high Ms to magnetize recording medium up to 1 Tadequate permeability at highy

•low noise•high permeability

adequate permeability at high frequency

g p y•low magnetostriciton•small

Recording medium:•should respond to field of write

•hard surface•inductive or

phead and retain information•coercivity 500 - 3000 Oe

magnetoresistance •suitable remanent magnetization•small, single domain particles (103 for a bit)(103 for a bit)